Geometry for Navigation

The principles used in navigation both on sea, land and in outer space. We look at trilateration and triangulation.

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A sextant is a device which measures the angle between two points. Using these angle and knowledge of geometry we can navigate.

Trigonometry Crash Course

Let us start with the most basic geometry. If we got a right triangle (one angle is 90°) and we know either:

  1. A length and an angle
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a² = b² + c²
sin(α) = b/a
cos(α) = c/a
tan(α) = b/c
sin(β) = c/a
cos(β) = b/a
tan(α) = c/b
x = cos(θ)
y = sin(θ)
w = r*cos(θ)
h = r*sin(θ)
tan(α) = c/b

Navigation on a Flat World

Lets avoid complicating things, and consider the cases when the earth is flat and the sun is relatively close. A point directly below a heavenly object (a star, moon, sun or planet), is called a Geographical Position (GP). In this example A is GP. Lets say we got a person at some position B, who doesn’t know where he is. He tries to figure it out by measuring the angle θ to the sun at his position.

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r = h/tan(α)

Far Away Objects

A common issue when dealing with navigation is that we try to find angles to objects extremely far away, such as stars or the sun.

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Navigation on a Curve World

In the animation below we assume a curved world, with a sun so far away that the sun lights hit the surface with the same parallel lines.

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Spherical World

Of course in reality the earth is spherical. If we measure an angle to a distant object, we are not on some point along a line but rather along a circle, as you can see below.

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Trilateration

Of course we would not be happy simply knowing which circle were are on. We want to know a single point. To accomplish that we use triangulation. Or more specifically when we are dealing with circles we call it trilateration.

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d = h*tan(α)
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Triangulation

When we find our position by measuring angles rather than distances (radius of a circle) we call that triangulation. We can e.g. look at a lighthouse using a bearing compass to determine how many degrees relative to magnetic north it is. A slightly more complicated case is to find the angle between two known objects. That would be required if e.g. we don’t have a magnetic north, or we don’t have a compass. Other planets such as Mars of Venus don’t have magnetic norths.

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Geek dad, living in Oslo, Norway with passion for UX, Julia programming, science, teaching, reading and writing.

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